Adaptive control based on retrospective cost optimization

ABSTRACT

A discrete-time adaptive control law for stabilization, command following, and disturbance rejection that is effective for systems that are unstable, MIMO, and/or nonminimum phase. The adaptive control algorithm includes guidelines concerning the modeling information needed for implementation. This information includes the relative degree, the first nonzero Markov parameter, and the nonminimum-phase zeros. Except when the plant has nonminimum-phase zeros whose absolute value is less than the plant&#39;s spectral radius, the required zero information can be approximated by a sufficient number of Markov parameters. No additional information about the poles or zeros need be known. Numerical examples are presented to illustrate the algorithm&#39;s effectiveness in handling systems with errors in the required modeling data, unknown latency, sensor noise, and saturation.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.61/201,035 filed on Dec. 5, 2008. The entire disclosure of the aboveapplication is incorporated herein by reference.

This application is also related to U.S. Pat. No. 6,208,739, issued onMar. 27, 2001, which is incorporated herein by reference.

GOVERNMENT SUPPORT

This invention was made with government support under Grant No.NNX08AB92A awarded by NASA. The government has certain rights in theinvention.

FIELD

The present disclosure relates to methods and control systems for usinga digital adaptive control algorithm and, more particularly, relates tomethods and control systems for using a digital adaptive controlalgorithm that are based on a retrospective correction feedback filter.

BACKGROUND AND SUMMARY

This section provides background information related to the presentdisclosure which is not necessarily prior art. This section alsoprovides a general summary of the disclosure, and is not a comprehensivedisclosure of its full scope or all of its features.

Unlike robust control, which chooses control gains based on a prior,fixed level of modeling uncertainty, adaptive control algorithms tunethe feedback gains in response to the true dynamical system (or“plant”), and commands and disturbances (collectively “exogenoussignals”). Generally speaking, adaptive controllers require less priormodeling information than robust controllers, and thus can be viewed ashighly parameter-robust control laws. The price paid for the ability ofadaptive control laws to operate with limited prior modeling informationis the complexity of analyzing and quantifying the stability andperformance of the closed-loop system, especially in light of the factthat adaptive control laws, even for linear plants, are nonlinear.

Stability and performance analysis of adaptive control laws oftenentails assumptions on the dynamics of the plant. For example, a widelyinvoked assumption in adaptive control is passivity, which isrestrictive and difficult to verify in practice. A related assumption isthat the plant is minimum phase, which may entail the same difficulties.In fact, sampling may give rise to nonminimum-phase zeros whether or notthe continuous-time system is minimum phase, which must ultimately beaccounted for by any adaptive control algorithm implemented digitally ina sampled-data control system. Beyond these assumptions, adaptivecontrol laws are known to be sensitive to unmodeled dynamics and sensornoise, which necessitates robust adaptive control laws.

In addition to these basic issues, adaptive control laws may entailunacceptable transients during adaptation, which may be exacerbated byactuator limitations. In fact, adaptive control under extremely limitedmodeling information, such as uncertainty in the sign of thehigh-frequency gain, may yield a transient response that exceeds thepractical limits of the plant. Therefore, the type and quality of theavailable modeling information as well as the speed of adaptation mustbe considered in the analysis and implementation of adaptive controllaws.

Adaptive control laws have been developed in both continuous-time anddiscrete-time settings. In the present application we considerdiscrete-time adaptive control laws since these control laws can beimplemented directly in embedded code for sampled-data control systemswithout requiring an intermediate discretization step that may entailloss of stability margins.

According to some prior art, references on discrete-time adaptivecontrol include a discrete-time adaptive control law with guaranteedstability developed under a minimum-phase assumption. Extensions basedon internal model control and Lyapunov analysis also invoke thisassumption. To circumvent the minimum-phase assumption, the zeroannihilation periodic control law uses lifting to move all of the plantzeros to the origin. The drawback of lifting, however, is the need foropen-loop operation during alternating data windows. An alternativeapproach, is to exploit knowledge of the nonminimum-phase zeros.Knowledge of the nonminimum-phase zeros is used to allow matching of adesired closed-loop transfer function, recognizing that minimum-phasezeros can be canceled but not moved, whereas nonminimum-phase zeros canneither be canceled nor moved. Knowledge of a diagonal matrix thatcontains the nonminimum-phase zeros is used within a MIMO directadaptive control algorithm. Finally, knowledge of the unstable zeros ofa rapidly sampled continuous-time SISO system with a realnonminimum-phase zero is used in some instances.

Motivated by the adaptive control laws given in some instances, the goalof the present application is to develop a discrete-time adaptivecontrol law that is effective for nonminimum-phase systems. Inparticular, we present an adaptive control algorithm that extends theretrospective cost optimization approach. This extension is based on aretrospective cost that includes control weighting as well as a learningrate, which can be used to adjust the rate of controller convergence andthus the transient behavior of the closed-loop system. Unlike someinstances, which use a gradient update, the present application uses aNewton-like update for the controller gains as the closed-form solutionto a quadratic optimization problem. No off-line calculations are neededto implement the algorithm or control system. A key aspect of thisextension is the fact that the required modeling information is therelative degree, the first nonzero Markov parameter, andnonminimum-phase zeros, if any. Except when the plant hasnonminimum-phase zeros whose absolute value is less than the plant'sspectral radius, we show that the required zero information can beapproximated by a sufficient number of Markov parameters from thecontrol inputs to the performance variables. No matching conditions arerequired on either the plant uncertainty or disturbances.

In some embodiments, a goal of the present application is to develop theRCF adaptive control algorithm and demonstrate its effectiveness forhandling nonminimum-phase zeros. To this end we consider a sequence ofexamples of increasing complexity, ranging from SISO, minimum-phaseplants to MIMO, nonminimum-phase plants, including stable and unstablecases. We then revisit these plants under off-nominal conditions, thatis, with uncertainty in the required plant modeling data, unknownlatency, sensor noise, and saturation. These numerical examples provideguidance into choosing the design parameters of the adaptive control lawin terms of the learning rate, data window size, controller order,modeling data, and control weightings.

According to the principles of the present teachings, a discrete-timeadaptive control law or algorithm for stabilization, command following,and disturbance rejection that is effective for systems that areunstable, MIMO, and/or nonminimum phase. The adaptive control algorithmincludes guidelines concerning the modeling information needed forimplementation. This information includes the relative degree, the firstnonzero Markov parameter, and the nonminimum-phase zeros. Except whenthe plant has nonminimum-phase zeros whose absolute value is less thanthe plant's spectral radius, the required zero information can beapproximated by a sufficient number of Markov parameters. No additionalinformation about the poles or zeros need be known. We present numericalexamples to illustrate the algorithm's effectiveness in handling systemswith errors in the required modeling data, unknown latency, sensornoise, and saturation.

Further areas of applicability will become apparent from the descriptionprovided herein. The description and specific examples in this summaryare intended for purposes of illustration only and are not intended tolimit the scope of the present disclosure.

DRAWINGS

The drawings described herein are for illustrative purposes only ofselected embodiments and not all possible implementations, and are notintended to limit the scope of the present disclosure.

FIG. 1 depicts a closed-loop system including adaptive control algorithmwith the retrospective correction filter (dashed box) for p=1.

FIG. 2 depicts roots of p₂₀(q) for the stable, non-minimum-phase plantin Example 5.1. The dashed line denotes ρ(A)=0.95. Note that the rootsoutside ρ(A) are close to the outer nonminimum-phase zeros −1.5 and1.25. The remaining roots are either located at the origin or form anapproximate ring with radius close to ρ(A).

FIG. 3 depicts roots of p₂₅(q) for the unstable, nonminimum-phase plantin Example 5.2. The dashed line denotes ρ(A)=1.4. Note that the root ofp₂₅(q) outside ρ(A) is close to the outer nonminimum-phase zero −1.5.However, the nonminimum-phase zero 1.25 is not approximated by a root ofp₂₅(q). The remaining roots are either located at the origin or form anapproximate ring with radius close to ρ(A).

FIG. 4 depicts roots of {tilde over (p)}₂₅(q) for the unstable,nonminimum-phase plant in Example 5.3. The dashed line denotesρ(Ã)=0.95, where Ã is the dynamics matrix of a minimal realization of{tilde over (G)}_(zu). Note that the roots outside ρ(Ã) are close to theinner and outer nonminimum-phase zeros of G_(zu). The remaining rootsare either located at the origin or form an approximate ring with radiusclose to ρ(Ã).

FIG. 5 depicts closed-loop response of the unstable, minimum-phase, SISOplant in Example 7.1 using the nonminimum-phase-zero-based constructionof B _(zu). The control is turned on at k=0. The controller order isn_(c)=2 with parameters p=1 and α(k)≡10.

FIG. 6 depicts closed-loop response of the unstable, minimum-phase, SISOplant in Example 7.2 with a step command. The control is turned on atk=200. The controller order is n_(c)=10 with parameters p=5, α(k)≡5, andr=10 with B _(zu) given by (56).

FIG. 7 depicts closed-loop response of the stable, minimum-phase, SISOplant in Example 7.1 with a step command and sinusoidal disturbance. Thecontrol is turned on at k=200. The controller order is n_(c)=20 withparameters p=1, α(k)≡50, and r=3 with B _(zu) given by (56).

FIG. 8 depicts closed-loop disturbance-rejection response of the stable,minimum-phase, SISO plant in Example 7.4. The control is turned on atk=200. The controller order is n_(c)=15 with parameters p=1, α(k)≡25,and r=3 with B _(zu) given by (56).

FIG. 9 depicts time history of the components of θ(k) for the stable,minimum-phase, SISO plant in Example 7.4. The control is turned on atk=200.

FIG. 10 depicts bode magnitude plot of the adaptive controller inExample 7.4 at k=1000 samples. The adaptive controller places poles atthe disturbance frequencies Ω₁=π/10 rad/sample and Ω₂13π/50 rad/sample.The controller magnitude |G_(c)(e^(jΩ))| is plotted for Ω up to theNyquist frequency Ω_(Nyq)=π rad/sample.

FIG. 11 depicts closed-loop disturbance-rejection response of thestable, nonminimum-phase, SISO plant in Example 7.5. The control isturned on at k=200. The controller order is n_(c)15 with parameters p=1,α(k)≡25, and r=7 with B _(zu) given by (56).

FIG. 12 depicts closed-loop disturbance-rejection response of thestable, nonminimum-phase, SISO plant in Example 7.5. The control isturned on at k=200. The controller order is n_(c)15 with parameters p=1,α(k)≡2500, and r=7 with B _(zu) given by (56). Compared to FIG. 11, theinitial transient is reduced at the expense of convergence speed.

FIG. 13 depicts closed-loop disturbance-rejection response of theunstable, minimum-phase, SISO plant in Example 7.6. The control isturned on at k=200. The controller order is n_(c)=15 with parametersp=1, α(k)≡25, and r=10 with B _(zu) given by (56).

FIG. 14 depicts closed-loop disturbance-rejection response of thestable, minimum-phase, two-input, two-output plant in Example 7.7. Thecontrol is turned on at k=200. The controller order is n_(c)=15 withparameters p=1, α(k)≡1, and r=10 with B _(zu) given by (56).

FIG. 15 depicts closed-loop disturbance-rejection response of thestable, nonminimum-phase, two-input, two-output plant in Example 7.8.The control is turned on at k=200. The controller order is n_(c)=15 withparameters p=2, α(k)≡1, and r=8 with B _(zu) given by (56).

FIG. 16 depicts closed-loop disturbance-rejection response of theunstable, nonminimum-phase, two-input, two-output plant in Example 7.9.The control is turned on at k=200. The controller order is n_(c)=10 withparameters p=1, α(k)≡1, and r=10 with B _(zu) given by (56).

FIG. 17 depicts closed-loop performance comparison of the stable,nonminimum-phase, SISO plant in Example 7.5 with multiplicative error inB. We take n_(c)=10, p=1, and α(k)≡1000. The multiplicative error η,which is used to obtain the Markov parameters for B _(zu) given by (56)with r=10, is varied between 0.3 and 5. The best performance is obtainedfor η=1, which corresponds to the true value of B.

FIG. 18 depicts closed-loop performance comparison of the stable,nonminimum-phase, SISO plant in Example 8.3 with a multiplicative errorin the nonminimum-phase zero 2. We take n_(c)=10, p=1, and α(k)≡25. Thenonminimum-phase-zero multiplicative error η, which is used to constructB _(zu) given by (52), is varied between 0.75 and 2.5. The bestperformance is obtained for η=1.05, which is close to the true value ofthe nonminimum-phase zero.

FIG. 19 depicts closed-loop response of the unstable, minimum-phase,SISO plant in Example 7.6 with random white noise added to themeasurement. The control is turned on at k=0. The controller order isn_(c)=15 with parameters p=1, α(k)≡25, and r=3 with B _(zu) given by(56). The performance variable is degraded to the level of the additivesensor noise v(k).

FIG. 20 depicts closed-loop disturbance-rejection response of thestable, minimum-phase, SISO plant in Example 7.4, where both theactuator and sensor are saturated at ±2. The control is turned on atk=200. The controller order is n_(c)=15 with parameters p=1, α(k)≡25,and r=3 with B _(zu) given by (56). The saturations degrade steady-stateperformance.

FIG. 21 depicts closed-loop step-command-following responses of thestable, minimum-phase, SISO plant in Example 7.4 with and withoutactuator saturation at ±0.1. The control is turned on at k=200. Thecontroller order is n_(c)=15 with parameters p=1, α(k)≡25, and r=3 withB _(zu) given by (56).

FIG. 22 depicts model reference adaptive control problem withperformance variable z.

FIG. 23 depicts closed-loop model reference adaptive control of Boeing747 longitudinal dynamics. The controller order is n_(c)=10 withparameters p=1, α(k)≡40, and r=10 with B _(zu) given by (56). Thecontroller is turned on at t=0 sec, and the performance variableconverges within about 20 sec.

FIG. 24 depicts closed-loop model reference adaptive control of missilelongitudinal dynamics. The control effectiveness λ=1, and thus the plantand reference model are identical. Therefore, the adaptive control inputu_(ac)=0.

FIG. 25 depicts missile longitudinal dynamics with control effectivenessλ=0.50 and adaptive controller turned off, that is, autopilot-onlycontrol.

FIG. 26 depicts closed-loop model reference adaptive control of missilelongitudinal dynamics with control effectiveness λ=0.50. The augmentedcontrollers provide better performance than the autopilot-onlysimulation.

FIG. 27 depicts closed-loop model reference adaptive control of missilelongitudinal dynamics with control effectiveness λ=0.25. After atransient, the augmented controllers stabilize the system, whereas theautopilot-only simulation fails. Note that the system is stabilizeddespite the total control input u reaching the actuator saturation levelof ±30 deg.

FIG. 28 depicts closed-loop model reference adaptive control of missilelongitudinal dynamics with control effectiveness λ=0.25. The adaptivecontroller is initialized with the converged gains from the 50% controleffectiveness case. The initial transient is reduced as compared withinitializing the control gains to zero. In this case, the actuatorsaturation level is never reached.

Corresponding reference numerals indicate corresponding parts throughoutthe several views of the drawings.

DETAILED DESCRIPTION

Example embodiments will now be described more fully with reference tothe accompanying drawings. Example embodiments are provided so that thisdisclosure will be thorough, and will fully convey the scope to thosewho are skilled in the art. Numerous specific details are set forth suchas examples of specific components, devices, and methods, to provide athorough understanding of embodiments of the present disclosure. It willbe apparent to those skilled in the art that specific details need notbe employed, that example embodiments may be embodied in many differentforms and that neither should be construed to limit the scope of thedisclosure.

The terminology used herein is for the purpose of describing particularexample embodiments only and is not intended to be limiting. As usedherein, the singular forms “a”, “an” and “the” may be intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. The terms “comprises,” “comprising,” “including,” and“having,” are inclusive and therefore specify the presence of statedfeatures, integers, steps, operations, elements, and/or components, butdo not preclude the presence or addition of one or more other features,integers, steps, operations, elements, components, and/or groupsthereof. The method steps, processes, and operations described hereinare not to be construed as necessarily requiring their performance inthe particular order discussed or illustrated, unless specificallyidentified as an order of performance. It is also to be understood thatadditional or alternative steps may be employed.

Problem Formulation

We begin by first considering the MIMO discrete-time system

x(k+1)=Ax(k)+Bu(k)+D ₁ w(k),  (1)

y(k)=Cx(k)+D ₂ w(k),  (2)

z(k)=E ₁ x(k)+E ₀ w(k),  (3)

where x(k)ε

^(n), y(k)ε

^(l) ^(y) , z(k)ε

^(l) ^(z) , u(k)ε

^(l) ^(u) , w(k)ε

^(l) ^(w) , and k≧0. Our goal is to develop an adaptive output feedbackcontroller under which the performance variable z is minimized in thepresence of the exogenous signal w. In (1)-(3), w can represent either acommand signal to be followed, an external disturbance to be rejected,or both. For example, if D₁=0 and E_(o)≠0, then the objective is to havethe output E₁x follow the command signal −E₀w. On the other hand, ifD₁≠0 and E_(o)=0, then the objective is to reject the disturbance w fromthe performance variable E₁x. The combined command-following anddisturbance-rejection problem is addressed when D₁ and E₀ are suitablypartitioned matrices. More precisely, if D₁=[D₁₁0], E₀=[0E₀₂], and

${{w(k)} = \begin{bmatrix}{w_{1}(k)} \\{w_{2}(k)}\end{bmatrix}},$

then the objective is to have E₁x follow the command −E₀₂w₂ whilerejecting the disturbance D₁₁w₁. Lastly, if D₁ and E₀ are zero matrices,then the objective is output stabilization, that is, convergence of z tozero. We assume that (A, B) is stabilizable, and (A, C) and (A, E₁) aredetectable, and that measurements of y and z are available for feedback.If the command signal is included as a component of y, then the adaptivecontroller has a feedforward architecture. For disturbance-rejectionproblems, the controller does not require measurements of the externaldisturbance w.

ARMAX Modeling

Consider the ARMAX representation of (1), (3), given by

$\begin{matrix}{{{z(k)} = {{\sum\limits_{i = 1}^{n}{{- \alpha_{i}}{z\left( {k - i} \right)}}} + {\sum\limits_{i = 1}^{n}{\beta_{i}{u\left( {k - i} \right)}}} + {\sum\limits_{i = 0}^{n}{\gamma_{i}{w\left( {k - i} \right)}}}}},} & (4)\end{matrix}$

where α₁, . . . , α_(n)ε

, β₁, . . . , β_(n)ε

^(l) ^(z) ^(×l) ^(u) , and γ₀, . . . , γ_(n)ε

^(l) ^(z) ^(×l) ^(w) . We define the relative degree d≧1 as the smallestpositive integer i such that the ith Markov parameter

$H_{i}\overset{\Delta}{=}{{E_{1}A^{i - 1}B} \in {\mathbb{R}}^{l_{z} \times l_{u}}}$

is nonzero. Note that, if d=1, then H₁=β₁, whereas, if d≧2, then β₁= . .. =β_(d−1)=H₁= . . . =H_(d−1)=0 and H_(d)=β_(d).

Letting the data window size p be a positive integer, we define theextended performance vector Z(k)ε

^(pl) ^(z) and U₁(k)ε

^(q) ^(c) ^(l) ^(u) by

$\begin{matrix}{{{Z(k)}\overset{\Delta}{=}\begin{bmatrix}{z(k)} \\\vdots \\{z\left( {k - p + 1} \right)}\end{bmatrix}},{{U_{1}(k)}\overset{\Delta}{=}\begin{bmatrix}{u\left( {k - 1} \right)} \\\vdots \\{u\left( {k - q_{c}} \right)}\end{bmatrix}},} & (5)\end{matrix}$

where

$q_{c}\overset{\Delta}{=}{n + p - 1.}$

The data window size p has a small but noticeable effect on transientbehavior. Now, (4) can be written in the form

Z(k)=W _(zw)φ_(zw)(k)+B _(zu) U ₁(k),  (6)

where W_(zw)ε

^(pl) ^(z) ^(×[q) ^(c) ^(l) ^(z) ^(+(q) ^(c) ^(+1)l) ^(w) ^(]), B_(zu)ε

^(pl) ^(z) ^(×q) ^(c) ^(l) ^(u) , and φ_(zw)ε

^(q) ^(c) ^(l) ^(z) ^(+(q) ^(c) ^(+1)l) ^(w) are given by

$\begin{matrix}{{W_{zw}\overset{\Delta}{=}\left\lbrack \begin{matrix}{{- \alpha_{1}}I_{l_{z}}} & \ldots & {{- \alpha_{n}}I_{l_{z}}} & 0_{l_{z} \times l_{z}} & \ldots & 0_{l_{z} \times l_{z}} & \gamma_{0} & \ldots & \gamma_{n} & 0_{l_{z} \times l_{w}} & \ldots & 0_{l_{z} \times l_{w}} \\0_{l_{z} \times l_{z}} & \ddots & \; & \ddots & \ddots & \vdots & 0_{l_{z} \times l_{w}} & \ddots & \; & \ddots & \ddots & \vdots \\\ddots & \ddots & \ddots & \; & \ddots & 0_{l_{z} \times l_{z}} & \vdots & \ddots & \ddots & \; & \ddots & 0_{l_{z} \times l_{w}} \\0_{l_{z} \times l_{z}} & \ldots & 0_{l_{z} \times l_{z}} & {{- \alpha_{1}}I_{l_{z}}} & \ldots & {{- \alpha_{n}}I_{l_{z}}} & 0_{l_{z} \times l_{w}} & \ldots & 0_{l_{z} \times l_{w}} & \gamma_{0} & \ldots & \gamma_{n}\end{matrix} \right\rbrack},} & (7) \\{{B_{zu}\overset{\Delta}{=}\begin{bmatrix}\beta_{1} & \ldots & \beta_{n} & 0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} \\0_{l_{z} \times l_{u}} & \ddots & \; & \ddots & \ddots & \vdots \\\vdots & \ddots & \ddots & \; & \ddots & 0_{l_{z} \times l_{u}} \\0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} & \beta_{1} & \ldots & \beta_{n}\end{bmatrix}},{and}} & (8) \\{{\varphi_{zw}(k)}\overset{\Delta}{=}{\begin{bmatrix}{z\left( {k - 1} \right)} \\\vdots \\{z\left( {k - p - n + 1} \right)} \\{w(k)} \\\vdots \\{w\left( {k - p - n + 1} \right)}\end{bmatrix}.}} & (9)\end{matrix}$

Note that W_(zw) includes modeling information about the plant poles andexogenous input path, whereas B_(zu) includes modeling information aboutthe plant zeros. Both W_(zw) and B_(zu) have block-Toeplitz structure.

Controller Construction

To formulate an adaptive control algorithm for (1)-(3), we use astrictly proper time-series controller of order n_(c) such that thecontrol u(k) is given by

$\begin{matrix}{{{u(k)} = {{\sum\limits_{i = 1}^{n_{c}}{{P_{i}(k)}{u\left( {k - i} \right)}}} + {\sum\limits_{i = 1}^{n_{c}}{{Q_{i}(k)}{y\left( {k - i} \right)}}}}},} & (10)\end{matrix}$

where, for all i=1, . . . , n_(c), P_(i)(k)ε

^(l) ^(u) ^(×l) ^(u) and Q_(i)(k)ε

^(l) ^(u) ^(×l) ^(y) . The controller order n_(c) is determined bystandard control guidelines in terms of stabilization and disturbancerejection. The control (10) can be expressed as

$\begin{matrix}{{{u(k)} = {{\theta (k)}{\varphi (k)}}},{where}} & (11) \\{{\theta (k)}\overset{\Delta}{=}{\left\lbrack {{Q_{1}(k)}\mspace{14mu} \ldots \mspace{14mu} {Q_{n_{c}}(k)}{P_{1}(k)}\mspace{14mu} \ldots \mspace{14mu} {P_{n_{c}}(k)}} \right\rbrack \in {\mathbb{R}}^{l_{u} \times {n_{c}{({l_{u} + l_{y}})}}}}} & (12)\end{matrix}$

is the controller gain matrix, and the regressor vector φ(k) is given by

$\begin{matrix}{{\varphi (k)}\overset{\Delta}{=}{\begin{bmatrix}{y\left( {k - 1} \right)} \\\vdots \\{y\left( {k - n_{c}} \right)} \\{u\left( {k - 1} \right)} \\\vdots \\{u\left( {k - n_{c}} \right)}\end{bmatrix} \in {{\mathbb{R}}^{n_{c}{({l_{u} + l_{y}})}}.}}} & (13)\end{matrix}$

We define the extended control vector U(k)ε

^(p) ^(c) ^(l) ^(u) by

$\begin{matrix}{{{U(k)}\overset{\Delta}{=}\begin{bmatrix}{u\left( {k - 1} \right)} \\\vdots \\{u\left( {k - p_{c}} \right)}\end{bmatrix}},} & (14)\end{matrix}$

where p_(c)≧q_(c). Note that, if p_(c)=q_(c), then U(k)=U₁(k). From(11), it follows that the extended control vector U(k) can be written as

$\begin{matrix}{{{U(k)} = {\sum\limits_{i = 1}^{p_{c}}{L_{i}{\theta \left( {k - i} \right)}{\varphi \left( {k - i} \right)}}}},{where}} & (15) \\{L_{i}\overset{\Delta}{=}{\begin{bmatrix}0_{{({i - 1})}l_{u} \times l_{u}} \\I_{l_{u}} \\0_{{({p_{c} - i})}l_{u} \times l_{u}}\end{bmatrix} \in {{\mathbb{R}}^{{p_{c}l_{u}} + l_{u}}.}}} & (16)\end{matrix}$

Next, we define the retrospective performance vector {circumflex over(Z)}({circumflex over (θ)},k)ε

^(pl) ^(z) by

$\begin{matrix}{{{\hat{Z}\left( {\hat{\theta},k} \right)}\overset{\Delta}{=}{{W_{zw}{\varphi_{zw}(k)}} + {B_{zu}{U_{1}(k)}} - {{\overset{\_}{B}}_{zu}\left\lbrack {{U(k)} - {\hat{U}\left( {\hat{\theta},k} \right)}} \right\rbrack}}},} & (17)\end{matrix}$

where {circumflex over (θ)}ε

^(l) ^(u) ^(×n) ^(c) ^((l) ^(u) ^(+l) ^(y) ⁾ is the surrogate controllergain matrix, B _(zu) ε

^(pl) ^(z) ^(×p) ^(c) ^(l) ^(u) is the surrogate input matrix, and

$\begin{matrix}{{\hat{U}\left( {\hat{\theta},k} \right)}\overset{\Delta}{=}{\sum\limits_{i = 1}^{p_{c}}{L_{i}\hat{\theta}{\varphi \left( {k - i} \right)}}}} & (18)\end{matrix}$

is the recomputed extended control vector. Substituting (6) into (17)yields

{circumflex over (Z)}({circumflex over (θ)},k)=Z(k)− B _(zu)[U(k)={circumflex over (U)}({circumflex over (θ)},k)].  (19)

Note that the expression for {circumflex over (Z)}({circumflex over(θ)},k) given by (19) does not depend on either the exogenous signal wor the matrix W_(zw), which includes information about the open-looppoles as well as the transfer function from w to z. Hence, we do notneed to know this model data, and, when w represents a disturbance, wedo not need to assume that w is known. However, when w represents acommand, then w can be viewed as an additional measurement y, and thusthe controller has feedforward action. The matrix B _(zu) is discussedfurther below.

Note that (19) can be rewritten as

$\begin{matrix}{{{\hat{Z}\left( {\hat{\theta},k} \right)} = {{f(k)} + {{D(k)}{vec}\; \hat{\theta}}}},{where}} & (20) \\{{{f(k)}\overset{\Delta}{=}{{{Z(k)} - {{\overset{\_}{B}}_{zu}{U(k)}}} \in {\mathbb{R}}^{{pl}_{z}}}},} & (21) \\{{D(k)}\overset{\Delta}{=}{{\sum\limits_{i = 1}^{p_{c}}{{\varphi^{T}\left( {k - i} \right)} \otimes \left( {{\overset{\_}{B}}_{zu}L} \right)_{i}}} \in {\mathbb{R}}^{{pl}_{z}{xn}_{c}{l_{u}{({l_{u} + l_{y}})}}}}} & (22)\end{matrix}$

vec is the column-stacking operator, and

represents the Kronecker product.

Now, consider the retrospective cost function

$\begin{matrix}{{J\left( {\hat{\theta},k} \right)}\overset{\Delta}{=}{{{{\hat{Z}}^{T}\left( {\hat{\theta},k} \right)}{R_{1}(k)}{\hat{Z}\left( {\hat{\theta},k} \right)}} + {2\; {{\hat{Z}}^{T}\left( {\hat{\theta},k} \right)}{R_{12}(k)}{\hat{u}\left( {\hat{\theta},{k + 1}} \right)}} + {{{\hat{u}}^{T}\left( {\hat{\theta},{k + 1}} \right)}{R_{2}(k)}{\hat{u}\left( {\hat{\theta},{k + 1}} \right)}} + {{tr}\left\lbrack {{R_{3}(k)}\left( {\hat{\theta} - {\theta (k)}} \right)^{T}{R_{4}(k)}\left( {\hat{\theta} - {\theta (k)}} \right)} \right\rbrack}}} & (23)\end{matrix}$

where R₁(k)ε

^(pl) ^(z) ^(×pl) ^(z) , R₁₂(k)ε

^(pl) ^(z) ^(×l) ^(u) , R₂(k)ε

^(l) ^(u) ^(×l) ^(u) , R₃(k)ε

^(n) ^(c) ^((l) ^(u) ^(+l) ^(y) ^()×n) ^(c) ^((l) ^(u) ^(+l) ^(y) ⁾,R₄(k)ε

^(l) ^(u) ^(×l) ^(u) ,

$\left\lbrack \left. \quad\begin{matrix}{R_{1}(k)} & {R_{12}(k)} \\{R_{12}^{T}(k)} & {R_{2}(k)}\end{matrix} \right\rbrack \right.$

is positive semidefinite, R₃(k) and R₄(k) are positive definite, and

$\begin{matrix}{{\hat{u}\left( {\hat{\theta},k} \right)}\overset{\Delta}{=}{\hat{\theta}{{\varphi (k)}.}}} & (24)\end{matrix}$

Substituting (20) into (23) yields

$\begin{matrix}{{{J\left( {\hat{\theta},k} \right)} = {{\left( {{vec}\; \hat{\theta}} \right)^{T}{M(k)}{vec}\; \hat{\theta}} + {{b^{T}(k)}{vec}\; \hat{\theta}} + {c(k)}}}{where}} & (25) \\{{M(k)}\overset{\Delta}{=}{{{D^{T}(k)}{R_{1}(k)}{D(k)}} + {2\; {{D^{T}(k)}\left\lbrack {{\varphi^{T}(k)} \otimes {R_{12}(k)}} \right\rbrack}} + {\left\lbrack {{\varphi (k)}{\varphi^{T}(k)}} \right\rbrack \otimes {R_{2}(k)}} + {{R_{3}(k)} \otimes {R_{4}(k)}}}} & (26) \\{{b(k)}\overset{\Delta}{=}{{2{D^{T}(k)}{R_{1}(k)}{f(k)}} + {{2\left\lbrack {{\varphi (k)} \otimes {R_{12}^{T}(k)}} \right\rbrack}{f(k)}} - {{2\left\lbrack {{R_{3}(k)} \otimes {R_{4}(k)}} \right\rbrack}{vec}\; {\theta (k)}}}} & (27) \\{{c(k)}\overset{\Delta}{=}{{{f^{T}(k)}{R_{1}(k)}{f(k)}} + {{tr}\left\lbrack {{R_{3}(k)}{\theta^{T}(k)}{R_{4}(k)}{\theta (k)}} \right\rbrack}}} & (28)\end{matrix}$

Since M(k) is positive definite, J({circumflex over (θ)},k) has thestrict global minimizer θ(k+1) given by

$\begin{matrix}{{\theta \left( {k + 1} \right)} = {{- \frac{1}{2}}{{vec}^{- 1}\left\lbrack {{M^{- 1}(k)}{b(k)}} \right\rbrack}}} & (29)\end{matrix}$

Equation (29) is the adaptive control update law. Note that B _(zu)(which appears in f(k) and D(k)) must be specified in order to implement(29). Furthermore, (29) requires the on-line inversion of apositive-definite matrix of sizen_(c)l_(u)(l_(u)+l_(y))×n_(c)l_(u)(l_(u)+l_(y)).

In the special case

$\begin{matrix}{{{R_{1}(k)}\overset{\Delta}{=}I_{{pl}_{z}}},{{R_{12}(k)}\overset{\Delta}{=}0_{{pl}_{z} \times l_{u}}},{{R_{2}(k)}\overset{\Delta}{=}0_{l_{u} \times l_{u}}},} & (30) \\{{{R_{3}(k)}\overset{\Delta}{=}{{\alpha (k)}I_{n_{c}{({l_{u} + l_{y}})}}}},{{R_{4}(4)}\overset{\Delta}{=}I_{l_{u}}},} & (31)\end{matrix}$

where α(k)>0 is a scalar, (26)-(28) become

M(k)=D ^(T)(k)D(k)+α(k)I _(n) _(c) _(l) _(u) _((l) _(u) _(+l) _(y)₎,  (32)

b(k)=2D ^(T)(k)f(k)−2α(k)vecθ(k),  (33)

c(k)=f ^(T)(k)f(k)+α(k)tr[θ ^(T)(k)θ(k)]  (34)

Using the matrix inversion lemma it follows that

M ⁻¹(k)=α⁻¹(k)I _(n) _(c) _(l) _(u) _((l) _(u) _(+l) _(y) ₎−α⁻¹(k)D^(T)(k)[α(k)I _(pl) _(z) +D(k)D ^(T)(k)]⁻¹ D(k).  (35)

Consequently, in this case, the update law (29) requires the on-lineinversion of a positive-definite matrix of size pl_(z)×pl_(z). We usethe weightings (30), (31) for all of the examples in the presentapplication. The weighting parameter α(k) introduced in (31) is calledthe learning rate since it affects the convergence speed of the adaptivecontrol algorithm. As α(k) is increased, a higher weight is placed onthe difference between the previous controller coefficients and theupdated controller coefficients, and, as a result, convergence speed islowered. Likewise, as α(k) is decreased, convergence speed is raised. Byvarying α(k), we can effect tradeoffs between transient performance andconvergence speed.

We define the retrospective performance variable {circumflex over (z)}ε

^(l) ^(z) by

$\begin{matrix}{{\hat{z}(k)}\overset{\Delta}{=}{\left\lbrack {I_{l_{z}}\mspace{14mu} 0_{l_{z} \times l_{z}}\mspace{14mu} \ldots \mspace{14mu} 0_{l_{z} \times l_{z}}} \right\rbrack {{\hat{Z}\left( {{\theta (k)},k} \right)}.}}} & (36)\end{matrix}$

In the particular case z=y, using {circumflex over (z)} in place of y inthe regressor vector (13) yields faster convergence. Therefore, for z=y,we redefine (13) as

$\begin{matrix}{{\varphi (k)}\overset{\Delta}{=}{\begin{bmatrix}{\hat{z}\left( {k - 1} \right)} \\\vdots \\{\hat{z}\left( {k - n_{c}} \right)} \\{u\left( {k - 1} \right)} \\\vdots \\{u\left( {k - n_{c}} \right)}\end{bmatrix}.}} & (37)\end{matrix}$

The novel feature of the adaptive control algorithm given by (11) and(29) is the use of the retrospective correction filter (RCF) (19), asshown in FIG. 1 for p=1. RCF provides an inner loop to the adaptivecontrol law by modifying the extended performance vector Z(k) in termsof the difference between the actual past control inputs U(k) and therecomputed control inputs Û({circumflex over (θ)},k).

Markov-Parameter Polynomial

By recursively substituting (1) into (3), it follows that z(k) can berepresented by

z(k)=E ₁ A ^(r) x(k−r)+H ₁ u(k−1)+H ₂ u(k−2)+ . . . +H _(r) u(k−r)+H_(zw,0) w(k)+H _(zw,1) w(k−1)+ . . . +H _(zw,r) w(k−r),  (38)

where r≧d, H_(zw,0)

E₀, and, for all i>0, H_(zw,i)

E₁A^(i−1)D₁. In terms of the backward-shift operator q⁻¹, (38) can berewritten as

z(k)=E ₁ A ^(r) q ^(−r) x(k)+[H ₁ q ⁻¹ +H ₂ q ⁻² + . . . +H _(r) q ^(−r)]u(k)+[H _(zw,0) +H _(zw,1) q ⁻¹ + . . . +H _(zw,r) q ^(−r) ]w(k)  (39)

Shifting (39) forward by r steps gives

z(k+r)=E ₁ A ^(r) x(k)+p _(r)(q)u(k)+W _(r)(q)w(k),  (40)

where q is the forward-shift operator,

W_(r)(q)

H_(zw,0)q^(r)+H_(zw,1)q^(r−1)+H_(zw,2)q^(r−2)+ . . . +H_(zw,r),  (41)

and

p_(r)(q)

H₁q^(r−1)+H₂q^(r−2)+ . . . +H_(r).  (42)

We call p_(r)(q) the Markov-parameter polynomial. Note that p_(r)(q) isa matrix polynomial in the MIMO case and a polynomial in the SISO case.Furthermore, since H₁= . . . =H_(d−1)=0 when d≧2, it follows that, forall r≧d≧1, p_(r)(q) can be written as

p _(r)(q)=H _(d) q ^(r-d) +H _(d+1) q ^(r-d−1) + . . . +H _(r).  (43)

The Markov-parameter polynomial p_(r)(q) contains information about therelative degree d and, in the SISO case, the sign of the high-frequencygain, that is, the sign of H_(d). We show below that p_(r)(q) alsocontains information about the transmission zeros of G_(zu)(z)

E₁(zI−A)⁻¹B, which is given by

$\begin{matrix}{{G_{zu}(z)} = {\frac{1}{z^{n} + {\alpha_{1}z^{n - 1}} + \ldots + \alpha_{n}}\left( {{\beta_{1}z^{n - 1}} + {\beta_{2}z^{n - 2}} + \ldots + \beta_{n}} \right)}} & (44)\end{matrix}$

In order to relate the transmission zeros of G_(zu) to p_(r)(q), theLaurent series expansion of G_(zu) about z=∞ is given by

$\begin{matrix}{{G_{zu}(z)} = {\sum\limits_{i = 1}^{\infty}{z^{- i}{H_{i}.}}}} & (45)\end{matrix}$

This expansion converges uniformly on all compact subsets of{z:|z|>ρ(A)}, where ρ(A) is the spectral radius of A. By truncating thesummation in (45), we obtain the truncated Laurent expansion G _(r,zu)of G_(zu), given by

$\begin{matrix}\begin{matrix}{{{\overset{\_}{G}}_{r,{zu}}(z)}\overset{\Delta}{=}{\sum\limits_{i = 1}^{r}{z^{- i}H_{i}}}} \\{= {\frac{1}{z^{r}}\left( {{H_{1}z^{r - 1}} + \ldots + {H_{r - 1}z} + H_{r}} \right)}} \\{= {\frac{1}{z^{r}}{p_{r}(z)}}}\end{matrix} & (46)\end{matrix}$

Consequently, the Markov-parameter polynomial p_(r)(q) is closelyrelated to the truncated Laurent expansion of G_(zu).

Approximation of Outer Nonminimum-Phase Zeros

In the case of MIMO systems, p_(r)(q) is a matrix polynomial and thusdoes not have roots in the sense of a polynomial. We therefore requirethe notion of a Smith zero. Specifically, zε

is a Smith zero of p_(r)(q) if the rank of p_(r)(z) is less than thenormal rank of p_(r)(q), that is, the maximum rank of p_(r)(ξ) takenover all ξε

.

Definition 5.1—Let ξε

be a transmission zero of G_(zu). Then, ξ is an outer zero of G_(zu) if|ξ|≧ρ(A). Otherwise, ξ is an inner zero of G_(zu).

The following result shows that the Smith zeros of the Markov-parameterpolynomial p_(r)(q) asymptotically approximate each outer transmissionzero of G_(zu).

Fact 5.1—Let ξε

be an outer transmission zero of G_(zu). For each r, let

_(r)

{ξ_(r,1), . . . , ξ_(r,m) _(r) } denote the set of Smith zeros ofp_(r)(q). Then, there exists a sequence {ξ_(r,i) _(r) }^(∞) _(r=1) thatconverges to ξ as r→∞.

The following specialization to SISO transfer functions shows that theroots of p_(r)(q) asymptotically approximate each outer zero of G_(zu).

Fact 5.2 Consider l_(u)=l_(z)=1, and let ξε

be an outer zero of G_(zu). For each r, let

_(r)

{ξ_(r,1), . . . , ξ_(r,r-d)} be the set of roots of p_(r)(q). Then,there exists a sequence {ξ_(r,i) _(r) }^(∞) _(r=1) that converges to ξas r→∞.

The following examples illustrate Fact 5.2 by showing that, as rincreases, roots of the Markov-parameter polynomial p_(r)(q), and hence,roots of the numerator of the truncated transfer function G _(r,zu),asymptotically approximate each outer nonminimum-phase zero of F_(zu).The remaining roots of p_(r)(q) are either located at the origin or forman approximate ring with radius close to ρ(A). These roots are spuriousand have no effect on the adaptive control algorithm.

Example 5.1 (SISO, nonminimum-phase, stable plant). Consider the plantG_(zu) with d=2, H₂=1, poles 0.5±0.5 j, −0.5±0.5 j, ±0.95, ±0.7 j,minimum-phase zeros 0.3±0.7 j, −0.7±0.3 j, and outer nonminimum-phasezeros 1.25, −1.5. Table 1 lists the approximated nonminimum-phase zerosobtained as roots of p_(r)(q) as a function of r. Note that as rincreases, the outer nonminimum-phase zeros are more closelyapproximated by the roots of p_(r)(q). See FIG. 2.

TABLE 1 Approximated nonminimum-phase zeros obtained as roots ofp_(r)(q) as a function of r for the stable, nonminimum-phase plant inExample 5.1. As r increases, the outer zeros are more accuratelymodeled. r roots_(nmp) (p_(r)(q)) 6 {0.944, −1.537} 8 {1.170, −1.502} 10{1.207, −1.498} 15 {1.240, −1.499} 20 {1.248, −1.500} 25 {1.250, −1.500}

Example 5.2 (SISO, nonminimum-phase, unstable plant). Consider the plantG_(zu) with d=2, H₂=1, poles 0.5±0.5 j, −0.5±0.5 j, ±0.7 j, −0.95, 1.4minimum-phase zeros 0.3±0.7 j, −0.7±0.3 j, outer nonminimum-phase zero−1.5, and inner nonminimum-phase zero 1.25. FIG. 3 shows the roots ofp₂₅(q). Note that the root of p₂₅(q) outside ρ(A) is close to the outernonminimum-phase zero −1.5. However, the inner nonminimum-phase zero1.25 is not approximated by a root of p₂₅(q). The remaining roots areeither located at the origin or form an approximate ring with radiusclose to ρ(A).

Approximation of Inner Nonminimum-Phase Zeros

Example 5.2 illustrates that the roots of p_(r)(q) approximate eachouter nonminimum-phase zero of G_(zu). However, inner nonminimum-phasezeros of G_(zu) are not approximated by roots of p_(r)(q). To overcomethis deficiency, we can use information about the plant's unstable polesto create a modified Markov-parameter polynomial {tilde over (p)}_(r)(q)whose roots approximate each nonminimum-phase zero of G_(zu). Forillustration, assume that the SISO plant G_(zu) has a unique unstablepole ζε

whose absolute value is greater than all other poles of G_(zu). Then, wedefine

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{G}}_{zu}(z)}\overset{\Delta}{=}{\frac{z - \zeta}{z}{G_{zu}(z)}}} \\{= {{G_{zu}(z)} - {\frac{\zeta}{z}{G_{zu}(z)}}}} \\{= {{\sum\limits_{i = d}^{\infty}{z^{- i}H_{i}}} - {\sum\limits_{i = d}^{\infty}{z^{- {({i + 1})}}\zeta \; H_{i}}}}} \\{= {\sum\limits_{i = d}^{\infty}{z^{- i}\left\lbrack {H_{i} - {\zeta \; H_{i - 1}}} \right\rbrack}}} \\{= {\sum\limits_{i = d}^{\infty}{z^{- i}{\overset{\sim}{H}}_{i}}}}\end{matrix} & (47)\end{matrix}$

where, for i=1, 2, . . . , {tilde over (H)}_(i)

H_(i)−ζH_(i−1) are the modified Markov parameters, and H₀=0. Byrepeating this operation for each unstable pole of G_(zu), the roots ofthe modified Markov-parameter polynomial

$\begin{matrix}{{{\overset{\sim}{p}}_{r}(q)}\overset{\Delta}{=}{{{\overset{\sim}{H}}_{d}q^{r - d}} + {{\overset{\sim}{H}}_{d + 1}q^{r - d - 1}} + \ldots + {\overset{\sim}{H}}_{r}}} & (48)\end{matrix}$

can approximate each nonminimum-phase zero of G_(zu). The followingexample illustrates this process.

Example 5.3 (Ex. 5.2 with pole information) Reconsider Example 5.2,where the inner nonminimum-phase zero 1.25 is not approximated by a rootof p_(r)(q). Using knowledge of the unstable pole 1.4 to construct{tilde over (p)}_(r)(q) given by (48), FIG. 4 shows the roots of {tildeover (p)}₂₅(q). Note that the roots outside ρ(Ã), where Ã is thedynamics matrix of a minimal realization of G_(zu), are close to thenonminimum-phase zeros of G_(zu). The remaining roots are either locatedat the origin or form an approximate ring with radius close to ρ(Ã).

Construction of B _(zu)

We present four constructions for B _(zu) based on the availablemodeling information.

B_(zu)-Based Construction

If B_(zu) given by (8) is known, then, with p_(c)=q_(c), B _(zu) can bechosen to be equal to B_(zu). In this case, U(k)=U₁(k), and (17) becomes

{circumflex over (Z)}({circumflex over (θ)},k)=W _(zw)φ_(zw)(k)+B _(zu)Û({circumflex over (θ)},k).  (49)

This construction of B _(zu) captures information about the relativedegree d, the first nonzero Markov parameter (since H_(d)=β_(d)), andexact values of all transmission zeros of G_(zu), that is, bothminimum-phase and nonminimum-phase transmission zeros.

Nonminimum-Phase-Zero-Based Construction

Consider l_(u)=l_(z)=1 and assume that H_(d) and the nonminimum-phasezeros of G_(zu) are known. Then we define the nonminimum-phase-zeropolynomial N(q) to be the polynomial whose roots are equal to thenonminimum-phase zeros of G_(zu), that is,

$\begin{matrix}{{{N(q)}\overset{\Delta}{=}{{H_{d}q^{m}} + {{\overset{\sim}{\beta}}_{1}q^{m - 1}} + \ldots + {\overset{\sim}{\beta}}_{m}}},} & (50)\end{matrix}$

where m≧0 is the number of nonminimum-phase zeros in G_(zu), and {tildeover (β)}₁, . . . , {tilde over (β)}_(m)ε

. If m=0, that is, G_(zu) is minimum phase, then N(q)=H_(d). Withp_(c)=q_(c), the nonminimum-phase-zero-based construction of B _(zu) isthus given by

$\begin{matrix}{{{\overset{\_}{B}}_{zu} = \begin{bmatrix}H_{1} & \ldots & H_{d} & {\overset{\sim}{\beta}}_{1} & \ldots & {\overset{\sim}{\beta}}_{m} & 0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} & 0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} \\0_{l_{z} \times l_{u}} & \ddots & \; & \ddots & \ddots & \; & \ddots & \ddots & \; & \ddots & \ddots & \vdots \\\vdots & \ddots & \ddots & \; & \ddots & \ddots & \; & \ddots & \ddots & \; & \ddots & \vdots \\0_{l_{z} \times l_{u}} & \ldots & 0_{l_{2} \times l_{u}} & H_{1} & \ldots & H_{d} & {\overset{\sim}{\beta}}_{1} & \ldots & {\overset{\sim}{\beta}}_{m} & 0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}}\end{bmatrix}},} & (51)\end{matrix}$

where H₁= . . . =H_(d−1)=0. This construction of B _(zu) capturesinformation about the relative degree d, the first nonzero Markovparameter, and exact values of all nonminimum-phase zeros of G_(zu). Inthe minimum-phase case, the only required modeling information is H_(d).This construction of B _(zu) can be extended to the MIMO case byreplacing each minimum-phase zero in the Smith-McMillan form of G_(zu)by a zero at z=0.r-MARKOV-Based Construction

Replacing k with k−1 in (4) and substituting the resulting relation backinto (4) yields a 2-MARKOV model. Repeating this procedure r−1 timesyields the r-MARKOV model of (1)-(3)

$\begin{matrix}{{{z(k)} = {{\sum\limits_{i = 1}^{n}{\alpha_{r,i}{z\left( {k - r - i + 1} \right)}}} + {\sum\limits_{i = d}^{r - 1}{H_{i}{u\left( {k - i} \right)}}} + {\sum\limits_{i = 1}^{n}{\beta_{r,i}{u\left( {k - r - i + 1} \right)}}} + {\sum\limits_{i = 0}^{r - 1}{H_{{zw},i}{w\left( {k - i} \right)}}} + {\sum\limits_{i = 1}^{n}{\gamma_{r,i}{w\left( {k - r - i + 1} \right)}}}}},} & (52)\end{matrix}$

where, for i=1, . . . , n, the coefficients α_(r,i)ε

, β_(r,i)ε

^(l) ^(z) ^(×l) ^(u) , and γ_(r,i)ε

^(l) ^(z) ^(×l) ^(w) are given by

$\begin{matrix}\begin{matrix}{{\alpha_{1,i}\overset{\Delta}{=}{- \alpha_{i}}},} & {{\beta_{1,i}\overset{\Delta}{=}\beta_{i}},} & {{\gamma_{1,i}\overset{\Delta}{=}\gamma_{i}},} \\{\mspace{25mu} \vdots} & {\mspace{31mu} \vdots} & {\mspace{31mu} \vdots} \\{{\alpha_{r,i}\overset{\Delta}{=}{{\alpha_{{r - 1},1}\alpha_{1,i}} + \alpha_{{r - 1},{i + 1}}}},} & {{\beta_{r,i}\overset{\Delta}{=}{{\alpha_{{r - 1},1}\beta_{1,i}} + \beta_{{r - 1},{i + 1}}}},} & {{\gamma_{r,i}\overset{\Delta}{=}{{\alpha_{{r - 1},1}\gamma_{1,i}} + \gamma_{{r - 1},{i + 1}}}},} \\{\mspace{25mu} \vdots} & {\mspace{31mu} \vdots} & {\mspace{31mu} \vdots} \\{{\alpha_{r,n}\overset{\Delta}{=}{\alpha_{{r - 1},1}\alpha_{1,n}}},} & {{\beta_{r,n}\overset{\Delta}{=}{\alpha_{{r - 1},1}\beta_{1,n}}},} & {\gamma_{r,n}\overset{\Delta}{=}{\alpha_{{r - 1},1}{\gamma_{1,n}.}}}\end{matrix} & (53)\end{matrix}$

Note that β_(r,1)=H_(r) and γ_(r,1)=H_(zw,r). We represent (52) with w=0as the r-MARKOV transfer function

$\begin{matrix}{{{G_{r,{zu}}(z)} = \frac{1}{z^{r + n - 1} + {\alpha_{r,1}z^{n - 1}} + \ldots + \alpha_{r,n}}}\left( {{H_{1}z^{r + n - 2}} + \ldots + {H_{r - 1}z^{n}} + {H_{r}z^{n - 1}} + {\beta_{r,2}z^{n - 2}} + \ldots + \beta_{r,n}} \right)} & (54)\end{matrix}$

The system representation (54) is nonminimal since its order is n+r−1,and thus (54) includes poles that are not present in the original model.Furthermore, note that the coefficients of the terms z^(n+r−2) throughz^(n) in the denominator are zero. These facts are irrelevant for thefollowing development. Using the numerator coefficients of (54), ther-MARKOV-based construction of B _(zu) with p_(c)=q_(c)+r−1 is given by

$\begin{matrix}{{\overset{\_}{B}}_{zu} = \begin{bmatrix}H_{1} & \ldots & H_{r} & \beta_{r,2} & \ldots & \beta_{r,n} & 0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} \\0_{l_{z} \times l_{u}} & \ddots & \; & \ddots & \ddots & \; & \ddots & \ddots & \vdots \\\vdots & \ddots & \ddots & \; & \ddots & \ddots & \; & \ddots & 0_{l_{z} \times l_{u}} \\0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} & H_{1} & \ldots & H_{r} & \beta_{r,2} & \ldots & \beta_{r,n}\end{bmatrix}} & (55)\end{matrix}$

This construction of B _(zu) captures information about the relativedegree d, the first nonzero Markov parameter, and exact values of alltransmission zeros of G_(zu), that is, both minimum-phase andnonminimum-phase transmission zeros.

Markov-Parameter-Based Construction

Using the numerator coefficients of (46), the Markov-parameter-basedconstruction of B _(zu) with p_(c)=q_(c)+r−1 is given by

$\begin{matrix}{{\overset{\_}{B}}_{zu} = {\begin{bmatrix}H_{1} & \ldots & H_{r} & 0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} & 0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} \\0_{l_{z} \times l_{u}} & \ddots & \; & \ddots & \ddots & \; & \ddots & \ddots & \vdots \\\vdots & \ddots & \ddots & \; & \ddots & \ddots & \; & \ddots & \vdots \\0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}} & H_{1} & \ldots & H_{r} & 0_{l_{z} \times l_{u}} & \ldots & 0_{l_{z} \times l_{u}}\end{bmatrix}.}} & (56)\end{matrix}$

The Markov parameters are the numerator coefficients of a truncatedLaurent series expansion of G_(zu) about z=∞. The Markov parameterscontain information about the relative degree d and, as shown by Fact5.2 for the SISO case, approximate values of all outer nonminimum-phasezeros of G_(zu). The advantage in using B _(zu) given by (56) ratherthan (55) is that β_(r,2), . . . , β_(r,n) need not be known. If,however, G_(zu) has inner nonminimum-phase zeros and the unstable polesof G_(zu) whose absolute values are greater than at least one innernonminimum-phase zero are known, then we can replace the Markovparameters H₁, . . . , H_(r) in (56) by the modified Markov parameters{tilde over (H)}₁, . . . , {tilde over (H)}_(r) given in (47). If thesepoles are not known, then B _(zu) can be chosen to be either B_(zu), thenonminimum-phase-zero form (51), or the r-MARKOV form (55).

Note that, if the order n of the system is known and 2n+1 Markovparameters are available, then a state-space model of the system can bereconstructed by using the eigensystem realization algorithm. However,the examples considered in sections below use substantially fewer Markovparameters.

NUMERICAL EXAMPLES Nominal Cases

We now present numerical examples to illustrate the response of the RCFadaptive control algorithm under nominal conditions. We consider asequence of examples of increasing complexity, ranging from SISO,minimum-phase plants to MIMO, nonminimum-phase plants, including stableand unstable cases. Each SISO example is constructed such that H_(d)=1.All examples assume y=z with φ(k) given by (37), and, in allsimulations, the adaptive controller gain matrix θ(k) is initialized tozero. Unless otherwise noted, all examples assume x(0)=0.

Example 7.1 (SISO, minimum-phase, unstable plant, stabilization)Consider the plant G_(zu) with d=1, poles 0, 1.5, and innernonminimum-phase zero −1.25. For stabilization, we take D₁ and E₀ to bezero matrices. Let B _(zu) be given by (51), which is constructed usingthe first nonzero Markov parameter H₁=1 and the location of thenonminimum-phase zero −1.25, that is, N(q)=q+1.25. We take n_(c)=2, p=1,and α(k)≡10. The closed-loop response is shown in FIG. 5 for x(0)=[0.10.4]^(T).

Example 7.2 (SISO, minimum-phase, unstable plant, command following).Consider the double integrator plant G_(zu) with d=3, poles 0.5±0.5 j,−0.5±0.5 j, 1, 1, and a minimum-phase zeros 0.3±0.7 j, 0.5. We considera command-following problem with step command w(k)=1. With the plantrealized in controllable canonical form, we take D₁=0 and E₀=−1. We taken_(c)=10, p=5, α(k)≡5, and r=10 with B _(zu) given by (56). Theclosed-loop response is shown in FIG. 6.

Example 7.3 (SISO, minimum-phase, stable plant, command following anddisturbance rejection). Consider the plant G_(zu) with d=3, poles0.5±0.5 j, −0.5±0.5 j, ±0.9, ±0.7 j, and minimum-phase zeros 0.3±0.7 j,0.7±0.3 j, 0.5. We consider a combined step-command-following anddisturbance-rejection problem with command w₁ and disturbance w₂ givenby

$\begin{matrix}{{{w(k)} = {\begin{bmatrix}\begin{matrix}{w_{1}(k)} \\{w_{2}(k)}\end{matrix} \\\;\end{bmatrix} = \begin{bmatrix}\begin{matrix}5 \\{\sin \; \Omega_{1}k}\end{matrix} \\\;\end{bmatrix}}},} & (57)\end{matrix}$

where Ω₁=π/10 rad/sample. With the plant realized in controllablecanonical form, we take

$D_{1} = {{\begin{bmatrix}0 & 0 \\0 & 1\end{bmatrix}\mspace{14mu} {and}\mspace{14mu} E_{0}} = {\begin{bmatrix}{- 1} & 0\end{bmatrix}.}}$

The disturbance, which is not matched, is assumed to be unknown, and thecommand signal is not used directly. We take n_(c)=20, p=1, α(k)≡50, andr=3 with B _(zu) given by (56). The closed-loop response is shown inFIG. 7.

The following examples are disturbance-rejection simulations, that is,E₀=0, with the unknown two-tone sinusoidal disturbance

$\begin{matrix}{{{w(k)} = \begin{bmatrix}{\sin \; \Omega_{1}k} \\{{- 1.5}\; \sin \; \Omega_{2}k}\end{bmatrix}},} & (58)\end{matrix}$

where Ω₁=π/10 rad/sample and Ω₂=13π/50 rad/sample. With each plantrealized in controllable canonical form, we take

${D_{1} = \begin{bmatrix}I_{2} \\0\end{bmatrix}},$

and, therefore, the disturbance is not matched.

Example 7.4 (SISO, minimum-phase, stable plant, disturbance rejection)Consider the plant G_(zu) with d=3, poles 0.5±0.5 j, −0.5±0.5 j, ±0.9,±0.7 j, and minimum-phase zeros 0.3±0.7 j, −0.7±0.3 j, 0.5. Takingn_(c)=15, p=1, α(k)≡25, and r=3 with B _(zu) given by (56), theclosed-loop response is shown in FIG. 8. The control algorithm converges(see FIG. 9) to an internal model controller with high gain at thedisturbance frequencies, as seen in FIG. 10.

Example 7.5 (SISO, nonminimum-phase, stable plant, disturbancerejection) Consider the plant G_(zu) with d=3, poles 0.5±0.5 j, −0.5±0.5j, ±0.9, ±0.7 j, minimum-phase zeros 0.3±0.7 j, −0.7±0.3 j, and outernonminimum-phase zero 2. We take n_(c)=15, p=1, r=7, and α(k)≡25. TheMarkov-parameter polynomial used to construct B _(zu) as in (56) isgiven by p₇(q)=q⁴−1.2q³−0.96q²−0.56q−0.75, with roots 0.01±0.71 j,−0.77, 1.94. Note that the root 1.94 approximates the zero 2. Theclosed-loop response is shown in FIG. 11.

To illustrate the effect of the learning rate α(k), the closed-loopresponse is shown in FIG. 12 for α(k)≡2500 and all other parametersunchanged. Note that, with α(k)≡2500, the initial transient is reducedat the expense of convergence speed.

Example 7.6 (SISO, minimum-phase, unstable plant, disturbancerejection). Consider the plant G_(zu) with d=3, poles 0.5±0.5 j,−0.5±0.5 j, ±1.04, 0.1±1.025 j, and minimum-phase zeros 0.3±0.7 j,−0.7±0.3 j, 0.5. We take n_(c)=15, p=1, α(k)≡25, and r=10 with B _(zu)given by (56). The closed-loop response is shown in FIG. 13.

Example 7.7 (MIMO, minimum-phase, stable plant, disturbance rejection).Consider the two-input, two-output plant

${{G_{zu}(z)} = \begin{bmatrix}\frac{z^{2} - {0.5\; z}}{D_{1}(z)} & \frac{z^{4} - {0.1z^{3}} - {0.22z^{2}} + {0.59z} - 0.29}{D_{1}(z)} \\\frac{z - 0.5}{D_{1}(z)} & \frac{z^{3} - {1.1z^{2}} + {0.88z} - 0.29}{D_{1}(z)}\end{bmatrix}},$

where

${{D_{1}(z)}\overset{\Delta}{=}{z^{5} + {0.1z^{4}} + {0.09z^{3}} - {0.401z^{2}} - {0.196z} - 0.2205}},\; {d = 1},{{{and}\mspace{14mu} H_{1}} = {\begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}.}}$

Consequently, G_(zu) has poles −0.5±0.5 j, 0.9, ±0.7 j, −0.5±0.5 j, 0.9,±0.7 j and minimum-phase transmission zeros 0.3±0.7 j, 0.5, 0.5. We taken_(c)=15, p=1, α(k)≡1, and r=10 with B _(zu) given by (56). Theclosed-loop response is shown in FIG. 14.

Example 7.8 (MIMO, nonminimum-phase, stable plant, disturbancerejection) Consider the two-input, two-output plant

${{G_{zu}(z)} = \begin{bmatrix}\frac{z^{2} - {0.5z}}{D_{1}(z)} & \frac{z^{2} - z - 2}{D_{2}(z)} \\\frac{z - 0.5}{D_{1}(z)} & \frac{z - 2}{D_{2}(z)}\end{bmatrix}},$

where D₁(z) is in given n Example 7.7, D₂(z)z³−0.2z²+0.34z+0.232, d=1,and

$H_{1} = {\begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}.}$

Consequently, G_(zu) has poles −0.5±0.5 j, 0.3±0.7 j, ±0.7 j, −0.4, 0.9,minimum-phase transmission zero 0.5, and outer nonminimum-phasetransmission zero 2. We take n_(c)=15, p=2, α(k)≡1, and r=8 with B _(zu)given by (56). The closed-loop response is shown in FIG. 15.

Example 7.9 (MIMO, nonminimum-phase, unstable plant, disturbancerejection) Consider the two-input, two-output plant

${{G_{zu}(z)} = \begin{bmatrix}\frac{z^{2} - {0.5z}}{D_{3}(z)} & \frac{z^{2} - z - 2}{D_{4}(z)} \\\frac{z - 0.5}{D_{3}(z)} & \frac{z - 2}{D_{4}(z)}\end{bmatrix}},$

where

${{D_{3}(z)}\overset{\Delta}{=}{z^{5} - {1.1z^{4}} + {1.731z^{3}} - {1.494z^{2}} + {0.608z} - 0.4679}},{{D_{4}(z)}\overset{\Delta}{=}{z^{3} + {1.4z^{2}} + {0.9z} + 0.2}},{d = 1},{{{and}\mspace{14mu} H_{1}} = {\begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}.}}$

Consequently, G_(zu) has poles −0.5±0.5 j, ±0.7 j, 0.1±1.025 j, −0.4,0.9, minimum-phase transmission zero 0.5, and outer nonminimum-phasetransmission zero 2. We take n_(c)=10, p=1, α(k)≡1, and r=10 with B_(zu) given by (56). The closed-loop response is shown in FIG. 16.

NUMERICAL EXAMPLES Off-Nominal Cases

We now revisit the numerical examples of the preceding section toillustrate the response of the RCF adaptive control algorithm underconditions of uncertainty in the relative degree and Markov parametersas well as measurement noise and actuator and sensor saturation. In eachexample, the adaptive controller gain matrix θ(k) is initialized tozero. Unless otherwise noted, all examples assume x(0)=0.

Example 8.1 (Ex. 7.5 with Markov-parameter multiplicative error)Reconsider Example 7.5 with Markov-parameter multiplicative error. Forcontroller implementation, we use the estimate {circumflex over (B)}

ηB, where ηε

is varied between 0.3 and 5. For i=1, . . . , r, the estimated Markovparameters Ĥ_(i)=CA^(i−1){circumflex over (B)} are used to construct B_(zu) given by (56). Taking n_(c)=15, p=1, r=10, and α(k)≡1000, theclosed-loop performance is compared in FIG. 17. In each case, thecontrol is turned on at k=0, and the performance metric is given by

$\begin{matrix}{{k_{0}\overset{\Delta}{=}{\min \left\{ {{k \geq 9}:{{\frac{1}{10}{\sum\limits_{i = 0}^{9}{{z\left( {k - i} \right)}}}} < 0.01}} \right\}}},} & (59)\end{matrix}$

that is, k₀ is the minimum time step k such that the average of{|z(k−i)|}_(i=0) ⁹ is less than 0.01. FIG. 17 shows that the bestperformance is obtained for η=1, which corresponds to the true value ofB. As η is decreased, convergence slows significantly.

In the case where the sign of the first nonzero Markov parameter (thesign of the high-frequency gain) is wrong, that is, Ĥ₃=−J₃, thesimulation fails. These simulations suggest that performance degradationdue to an unknown scaling of the Markov parameters provides a usefulmeasure of adaptive gain margin. These findings are consistent with theadaptive gain-margin results.

Example 8.2 (Ex. 7.5 with unknown latency). A known latency of l stepscan be accounted for by replacing d by d+l in the construction of B_(zu). However, we now assess the effect of unknown latency in Example7.5, which is equivalent to uncertainty in the relative degree d. Thesystem has relative degree d=3. For controller implementation, we usethe erroneous estimate {circumflex over (d)} of d and take n_(c)=15,p=1, α(k)≡1000, and r=10 with B _(zu) given by (56). Letting {circumflexover (d)} be either 2, 3, 4, 5, or 6, Table 2 compares both theperformance metric (59) and the maximum value of |z(k)| for eachestimate {circumflex over (d)} of d. In each case, the control is turnedon at k=0. The best performance is obtained for {circumflex over(d)}=d=3.

TABLE 2 Closed-loop performance comparison of the stable,nonminimum-phase, SISO plant in Example 7.5 with unknown latency.{circumflex over (d)} k₀ max | z(k) | 2 1870 12.3 3 531 9.4 4 847 8.5 54633 10.9 6 11660 3.2 × 10⁹ For controller implementation, we use theerroneous estimate {circumflex over (d)} of d and take n_(c) = 15, p =1, α(k) ≡ 1000, and r = 10 with B _(zu) given by (56). The bestperformance is obtained for {circumflex over (d)} = d = 3

These simulations show the sensitivity of the adaptive controller tounknown errors in the relative degree d, which provides a useful measureof adaptive phase margin.

Example 8.3 (Sensitivity to nonminimum-phase-zero uncertainty). Considerthe plant G_(zu) with d=1, H₁=1, poles 0, 0.5, and outernonminimum-phase zero 2. The plant is subject to disturbance w(k) givenby (58), and thus, with the plant realized in controllable canonicalform, we take D₁=I₂ and E₀=0. Furthermore, we assume y=z and let φ(k) begiven by (37). To illustrate the sensitivity of the adaptive controlalgorithm to knowledge of the nonminimum-phase zero, we let B _(zu) begiven by (51), which is constructed using the first nonzero Markovparameter H₁=1, the nonminimum-phase zero 2, and a multiplicative errorηε

, that is, N(q)=q−2η. We vary η between 0.75 and 2.5 with n_(c)=10, p=1,and α(k)≡25. A closed-loop performance comparison is shown in FIG. 18.In each case, the control is turned on at k=0, and the performancemetric is given by (59). The best performance is obtained for η=1.05,which is close to the true value of the nonminimum-phase zero. Note thatthe adaptive control algorithm is more robust to larger values of η thansmaller values.

Example 8.4 (Ex. 7.6 with stabilization and noisy measurements).Reconsider Example 7.6 with no commands or disturbances. Forstabilization, we take D₁ and E₀ to be zero matrices. To assess theperformance of the adaptive algorithm with added sensor noise, we modify(2) and (3) by

y(k)=z(k)=E ₁ x(k)+E ₀ w(k)+v(k),  (60)

where v(k)ε

^(l) ^(z) is Gaussian white noise with mean v=2 and standard deviationσ=0.1. We take n_(c)=15, p=1, α(k)≡25, and r=3 with B _(zu) given by(56). For the initial condition

x(0)=[=0.43 −1.67 0.13 0.29 −1.15 1.19 1.19 −0.04]^(T),

the closed-loop response is shown in FIG. 19.

Example 8.5 (Ex. 7.4 with actuator and sensor saturation). ReconsiderExample 7.4 with the additional assumption that both the control inputand sensor measurement are subject to saturation at ±2. We taken_(c)=15, p=1, α(k)≡25, and r=3 with B _(zu) given by (56). Theclosed-loop response shown in FIG. 20 indicates that the saturationsdegrade steady-state performance.

Example 8.6 (Ex. 7.4 with command following and actuator saturation).Reconsider Example 7.4 with step command given by w(k)=1. With the plantrealized in controllable canonical form, we take D₁=0 and E₀=−1. Takingn_(c)=15, p=1, α(k)≡25, and r=3 with B _(zu) given by (56), theclosed-loop responses are shown in FIG. 21 with and without actuatorsaturation at ±0.1. With actuator saturation, the performance variablereflects the capability of the saturated control.

Model Reference Adaptive Control

Model reference adaptive control (MRAC), as illustrated in FIG. 22, is aspecial case of (1)-(3), where

$z\overset{\Delta}{=}{y_{1} - y_{m}}$

is the difference between the measured output y₁ of the plant G and theoutput y_(m) of a reference model G_(m). For MRAC, the exogenous commandw is assumed to be available to the controller as an additionalmeasurement variable y₂. Unlike standard MRAC methods, retrospectivecost adaptive control does not depend on knowledge of the referencemodel G_(m).

We now present numerical examples to illustrate the response of the RCFadaptive control algorithm for model reference adaptive control (seeFIG. 22). Unless otherwise noted, the adaptive controller gain matrixθ(k) is initialized to zero.

Boeing 747 Longitudinal Dynamics

Consider the longitudinal dynamics of a Boeing 747 aircraft, linearizedabout steady flight at 40,000 ft and 774 ft/sec. The inputs to thedynamical system are taken to be elevator deflection and thrust, whilethe output is the pitch angle. The continuous-time equations of motionare thus given by

$\begin{matrix}{\left\lbrack \begin{matrix}\overset{.}{u} \\\overset{.}{w} \\\overset{.}{q} \\\overset{.}{\theta}\end{matrix} \right\rbrack = {{\left\lbrack \begin{matrix}{- 0.003} & 0.039 & 0 & {- 0.322} \\{- 0.065} & {- 0.319} & 7.74 & 0 \\0.020 & {- 0.101} & {- 0.429} & 0 \\0 & 0 & 1 & 0\end{matrix} \right\rbrack\left\lbrack \begin{matrix}u \\w \\q \\\theta\end{matrix} \right\rbrack} + {\quad{{\begin{bmatrix}0.010 & 1 \\{- 0.180} & {- 0.040} \\{- 1.160} & 0.598 \\0 & 0\end{bmatrix}\begin{bmatrix}\delta_{e} \\\delta_{T}\end{bmatrix}},}}}} & (61) \\{\mspace{79mu} {{y = {\begin{bmatrix}y_{1} \\y_{2}\end{bmatrix} = {{\begin{bmatrix}0 & 0 & 0 & 1 \\0 & 0 & 0 & 0\end{bmatrix}\begin{bmatrix}u \\w \\q \\\theta\end{bmatrix}} + {\begin{bmatrix}0 \\1\end{bmatrix}w}}}},}} & (62) \\{\mspace{79mu} {{z = {y_{1} - y_{m}}},}} & (63)\end{matrix}$

where w is the exogenous command and y_(m) is the output of thereference model

$\begin{matrix}{{G_{m}(s)} = {\frac{Y_{m}(s)}{W(s)} = {\frac{0.0131}{s^{2} + {0.16s} + 0.0131}.}}} & (64)\end{matrix}$

We discretize (61)-(64) using a zero-order hold and sampling timeT_(s)=0.01 sec. The reference command is taken to be a 1-deg stepcommand in pitch angle. The controller order is n_(c)=10 with parametersp=1, α(k)≡40, and r=10 with B _(zu) given by (56). The closed-loopresponse is shown in FIG. 23 for zero initial conditions.

Missile Longitudinal Dynamics

We now present numerical examples for MRAC of missile longitudinaldynamics under off-nominal or damage situations. The missilelongitudinal plant is derived from the short period approximation of thelongitudinal equations of motion, given by

$\begin{matrix}{{\overset{.}{x} = {{\begin{bmatrix}{- 1.064} & 1 \\290.26 & 0\end{bmatrix}x} + {{\lambda \begin{bmatrix}{- 0.25} \\{- 331.4}\end{bmatrix}}u}}},} & (65) \\{{{y = {{\begin{bmatrix}{- 123.34} & 0 \\0 & 1\end{bmatrix}x} + {{\lambda \begin{bmatrix}{- 13.51} \\0\end{bmatrix}}u}}},{where}}{{x\overset{\Delta}{=}\begin{bmatrix}\alpha \\q\end{bmatrix}},\mspace{14mu} {y\overset{\Delta}{=}\begin{bmatrix}A_{z} \\q\end{bmatrix}},}} & (66)\end{matrix}$

and λε(0,1] represents the control effectiveness. Nominally, λ=1.

The open-loop system (65), (66) is statically unstable. To overcome thisinstability, a classical three-loop autopilot is wrapped around thebasic missile longitudinal plant. The adaptive controller then augmentsthe closed-loop system to provide control in off-nominal cases, that is,when λ<1. The autopilot and adaptive controller inputs are denotedu_(ap) and u_(ac), respectively. Thus, the total control inputu=u_(ap)+u_(ac). The reference model G_(m) consists of the basic missilelongitudinal plant with λ=1 and the classical three-loop autopilot. Anactuator amplitude saturation of ±30 deg=±0.524 rad is included in themodel, but no actuator or sensor dynamics are included.

The goal is to have the missile follow a pitch acceleration command wconsisting of a 1-g amplitude 1-Hz square wave. The performance variablez is the difference between the measured pitch acceleration A_(z) andthe reference model pitch acceleration A_(z)*, that is,

$z\overset{\Delta}{=}{A_{z} - {A_{z}^{*}.}}$

The closed-loop response is shown in FIG. 24 for λ=1. Since the plantand reference model are identical in the nominal case, the adaptivecontrol input u_(ac)=0.

All of the following examples use zero initial conditions and the sameadaptive controller parameters. The adaptive controller is implementedat a sampling rate of 300 Hz. We take n_(c)=3, p=1, and r=20 with B_(zu) given by (56). A time varying learning rate α(k)=75k+1 is usedsuch that, initially, controller adaptation is fast, and, as performanceimproves, the adaptation slows. The learning rate is identical for eachsimulation. System identification using the observer/Kalman filteridentification (OKID) algorithm is used to obtain the 20 Markovparameters required for controller implementation. The offlineidentification procedure is performed with a nominal simulation (λ=1) byinjecting band-limited white noise at the adaptive controller inputu_(ac) and recording the performance variable z while the autopilot isin the loop. No external disturbances are assumed to be present duringthe identification procedure.

Example 9.1 (50% control effectiveness). Consider λ=0.50. FIG. 25 showssimulation results with the adaptive controller turned off, that is,autopilot-only control. Now, with the autopilot augmented by theadaptive controller, simulation results are shown in FIG. 26. After atransient, the augmented controllers provide better performance than theautopilot-only simulation.

Example 9.2 (25% control effectiveness). Consider λ=0.25. With theadaptive controller turned off, that is, autopilot-only control, thesimulation fails. With the autopilot augmented by the adaptivecontroller, simulation results are shown in FIG. 27. After a transient,the augmented controllers stabilize the system, whereas theautopilot-only simulation fails.

FIG. 27 shows that the total control input u reaches the actuatorsaturation level of ±30 deg. To reduce the initial transient, weinitialize the adaptive controller with the converged control gains θfrom the 50% control effectiveness case. As shown in FIG. 28, theinitial transient is reduced as compared with initializing the controlgains to zero. In this case, the actuator saturation level is notreached.

CONCLUSION

We presented the RCF adaptive control algorithm, system, and method anddemonstrated its effectiveness in handling nonminimum-phase zerosthrough numerical examples illustrating the response of the algorithmunder conditions of uncertainty in the relative degree and Markovparameters, measurement noise, and actuator and sensor saturations.Bursting was not observed in any of the simulations. We also suggestedmetrics that can serve as gain and phase margins for discrete-timeadaptive systems. Development of Lyapunov-based stability and robustnessanalysis of the RCF adaptive control algorithm as well as development ofa theoretical foundation for analyzing broadband disturbance-rejectionproperties of the controller is anticipated.

The foregoing description of the embodiments has been provided forpurposes of illustration and description. It is not intended to beexhaustive or to limit the invention. Individual elements or features ofa particular embodiment are generally not limited to that particularembodiment, but, where applicable, are interchangeable and can be usedin a selected embodiment, even if not specifically shown or described.The same may also be varied in many ways. Such variations are not to beregarded as a departure from the invention, and all such modificationsare intended to be included within the scope of the invention.

1. A method for adaptive control of nonminimum-phase systems, saidmethod comprising: receiving input data; distinguishing between saidinput data and modified data; determining control gains from said inputdata and estimates of nonminimum phase zeros using a modified quadraticcost criterion that uses a retrospective performance vector; calculatinga control signal based on said control gains, said input data, and saidretrospective performance vector; and outputting said control signal. 2.The method according to claim 1 wherein said determining control gainscomprises determining control gains using estimates of a relativedegree, a first nonzero Markov parameter, and nonminimum-phase zeros. 3.The method according to claim 1 wherein said determining control gainscomprises determining control gains using an explicit minimizer of saidmodified quadratic cost criterion to achieve a minimizer in a singleoperational step.
 4. The method according to claim 1 wherein saiddetermining control gains comprises determining control gains using aone-step learning penalty.
 5. The method according to claim 1 whereinsaid determining control gains from said input data and estimates ofnonminimum phase zeros using a modified quadratic cost criterioncomprises a disturbance rejection response.
 6. The method according toclaim 1 wherein said determining control gains from said input data andestimates of nonminimum phase zeros using a modified quadratic costcriterion comprises a command following response.
 7. The methodaccording to claim 1 wherein said determining control gains from saidinput data and estimates of nonminimum phase zeros using a modifiedquadratic cost criterion comprises a stabilization response.
 8. Themethod according to claim 1, wherein said determining control gainsfurther comprises adjusting a rate of convergence.
 9. A system foradaptive control of nonminimum-phase systems, said system comprising: adevice receiving input data; a device distinguishing between said inputdata and modified data; a controller determining control gains from saidinput data and estimates of nonminimum phase zeros using a modifiedquadratic cost criterion, said controller calculating a control signalbased on said control gains and said input data and outputting saidcontrol signal.
 10. The system according to claim 9 wherein saidcontroller determines control gains using a relative degree, a firstnonzero Markov parameter, and nonminimum-phase zeros.
 11. The systemaccording to claim 9 wherein said controller determines control gainsusing estimated Markov parameters to obtain estimates of a relativedegree, a first nonzero Markov parameter, and nonminimum-phase zeros.12. The system according to claim 9 wherein said controller determinescontrol gains using an explicit minimizer of said modified quadraticcost criterion to achieve a minimum in a single operational step. 13.The system according to claim 9 wherein said controller determinescontrol gains using a one-step learning penalty.
 14. The systemaccording to claim 9 wherein said controller employs a modifiedquadratic cost criterion to define a disturbance rejection response. 15.The system according to claim 9 wherein said controller employs amodified quadratic cost criterion to define a command followingresponse.
 16. The system according to claim 9 wherein said controlleremploys a modified quadratic cost criterion to define a stabilizationresponse.
 17. The system according to claim 9, wherein said controllercomprises a device for adjusting a rate of convergence.